FINDING: The Fibonacci sequence is generated by a 2×2 matrix whose eigenvalues are the golden ratio φ and its reciprocal -1/φ; the matrix powers encode substitution tiling inflation dynamics for Penrose and related aperiodic tilings. MATH: - Fibonacci matrix: \ (F = pmatrix 1 & 1 \\ 1 & 0 pmatrix \), eigenvalues \ (₁ = = 1+52 1. 618 \), \ (₂ = -1 = 1-52 -0. 618 \). - Diagonalization: \ (Fⁿ = P pmatrix ⁿ & 0 \\ 0 & (-^-1) ⁿ pmatrix P^-1 \), yielding Binet formula: \ (Fₙ = ⁿ - (-) ^-n5 \). - Substitution tiling inflation factor = φ (scale factor for Penrose kite/dart inflation). - Complex root of \ (x³ = x + 1 \) (plastic constant ≈ 1. 3247) generates new substitution tilings (Ed Pegg finding). CONNECTION: - Golden ratio φ = 1. 618, its reciprocal 0. 618, and φ² = 2. 618 appear as eigenvalues and inflation scalars. - Penrose tilin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Fri,) studied this question.